If the volumes of two cubes are in the ratio of 64 : 125, then what is the ratio of their total surface area ?

Options

A. `9 : 16`
B. `4 : 5`
C. `16 : 25`
D. `64 : 125`

Answer

To solve the problem, we need to find the ratio of the total surface area of two cubes given that their volumes are in the ratio of 64:125.

### Step-by-Step Solution:

1. **Understand the Volume of a Cube**:
The volume \( V \) of a cube with side length \( a \) is given by the formula:
\[ V = a^3 \]
For two cubes, let the side lengths be \( A \) and \( B \). Therefore, the volumes can be expressed as \( A^3 \) and \( B^3 \).

2. **Set Up the Volume Ratio**:
We are given that the volumes of the two cubes are in the ratio:
\[ \frac{A^3}{B^3} = \frac{64}{125} \]

3. **Take the Cube Root**:
To find the ratio of the side lengths \( A \) and \( B \), we take the cube root of both sides:
\[ \frac{A}{B} = \sqrt[3]{\frac{64}{125}} = \frac{\sqrt[3]{64}}{\sqrt[3]{125}} = \frac{4}{5} \]

4. **Calculate the Total Surface Area**:
The total surface area \( S \) of a cube is given by the formula:
\[ S = 6a^2 \]
For our cubes, the total surface areas will be:
\[ S_A = 6A^2 \quad \text{and} \quad S_B = 6B^2 \]

5. **Set Up the Surface Area Ratio**:
We need to find the ratio of the total surface areas:
\[ \frac{S_A}{S_B} = \frac{6A^2}{6B^2} = \frac{A^2}{B^2} \]

6. **Substitute the Ratio of Side Lengths**:
We already have \( \frac{A}{B} = \frac{4}{5} \). Now we square this ratio:
\[ \frac{A^2}{B^2} = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]

7. **Final Ratio of Total Surface Areas**:
Therefore, the ratio of the total surface areas of the two cubes is:
\[ \frac{S_A}{S_B} = \frac{16}{25} \]

### Conclusion:
The ratio of the total surface area of the two cubes is \( 16:25 \).